Perturbation analysis of transient population dynamics using matrix projection models

Non?stable populations exhibit short?term transient dynamics: size, growth and structure that are unlike predicted long?term asymptotic stable, stationary or equilibrium dynamics. Understanding transient dynamics of non?stable populations is important for designing effective population management strategies, predicting the responses of populations to environmental change or disturbance, and understanding population processes and life?history evolution in variable environments.Transient perturbation analyses are vital tools for achieving these aims. They assess how transient dynamics are affected by changes to vital rates, population structure, or underlying variables that affect these. These changes could be imposed deliberately by population managers, or driven by environmental variables. Methodological approaches to transient perturbation analysis are diverse, and different methods are suited to different applications: choosing a method to use may be challenging.Here, I review existing methods for prospective transient perturbation analysis, and identify a number of key considerations for ecologists when choosing a method. These include the approach taken in calculating the perturbation, the type of model being analysed, the perturbation structure, the population response of interest, nonlinear response to perturbation, standardization for asymptotic dynamics, the initial population structure, and the time frame of interest. I discuss these with reference to the application of transient perturbation analyses in both population management and comparative analysis.The diversity of transient perturbation analyses available means that existing approaches are applicable to a wide range of population management and comparative analysis scenarios. It is important, however, for ecologists using these methods to know exactly what is being measured. Despite a wealth of existing methods, I identify some areas that would benefit from further development.</p

A pace and shape perspective on fertility

Ageing is ubiquitous to all organisms, but ageing does not always mean senescence. Counter to most evolutionary theories of ageing, the patterns of mortality and reproduction may remain unchanged or improve with age, as well as deteriorate. Describing this diversity presents a challenge to eco?evolutionary demography. The pace–shape framework of mortality tackled this challenge to qualify and quantify orthogonal components of ageing patterns in mortality. Here, we extend this framework to fertility.Analogous to the logic of the mortality framework, we define a perspective, a framework and novel methods for the pace and shape of fertility. These distinguish between orthogonal components of time?scale (pace) and distribution (shape) of reproduction over adult life span.Our pace and shape framework mirrors that of mortality, through a shift of perspective from the mother giving birth, to the offspring being born. Our new measures overcome many problems associated with measuring natural fertility trajectories, have both a clear biological and mathematical interpretation, can be intuitively visualized and satisfy and extend important conditions of the pace–shape paradigm.A comprehensive framework of fertility pace–shape facilitates ecological and evolutionary research addressing interactions and trade?offs between components of birth and death patterns, across the whole tree of life. The burgeoning emergence of large comparative demographic data sources across wide environmental, geographical, temporal and phylogenetic ranges, combined with pace–shape measures, opens the door to comparative analyses of ageing which were never possible before.</p

Beyond sensitivity: nonlinear perturbation analysis of transient dynamics

1. Perturbation analyses of population models are integral to population management: such analyses evaluate how changes in vital rates of members of the population translate to changes in population dynamics. Sensitivity and elasticity analyses of long?term (asymptotic) growth are popular, but limited: they ignore short?term (transient) dynamics and provide a linear approximation to nonlinear perturbation curves.2. Population inertia measures how much larger or smaller a non?stable population becomes compared with an equivalent stable population, as a result of transient dynamics. We present formulae for the transfer function of population inertia, which describes nonlinear perturbation curves of transient population dynamics. The method comfortably fits into wider frameworks for analytical study of transient dynamics, and for perturbation analyses that use the transfer function approach.3. We use case studies to illustrate how the transfer function of population inertia may be used in population management. These show that strategies based solely on asymptotic perturbation analyses can cause undesirable transient dynamics and/or fail to exploit desirable transient dynamics. This highlights the importance of considering both transient and asymptotic population dynamics in population management.4. Our case studies also show a tendency towards marked nonlinearity in transient perturbation curves. We extend our method to measure sensitivity of population inertia and show that it often fails to capture dynamics resulting from perturbations typical of management scenarios.</p

popdemo: an R package for population demography using projection matrix analysis

Effective population management requires accurate predictions of future population dynamics and how they may be manipulated to achieve management goals.2. The R package popdemo provides software tools for novel analytical methods that aim to enhance the predictive power of basic population projection matrix models. These include indices of transient population dynamics and transfer function analyses.3. We use a case study to demonstrate the use and importance of these methods for population management and briefly discuss their potential application outside population ecology.</p

A framework for studying transient dynamics of population projection matrix models

Empirical models are central to effective conservation and population management, and should be predictive of real?world dynamics. Available modelling methods are diverse, but analysis usually focuses on long?term dynamics that are unable to describe the complicated short?term time series that can arise even from simple models following ecological disturbances or perturbations. Recent interest in such transient dynamics has led to diverse methodologies for their quantification in density?independent, time?invariant population projection matrix (PPM) models, but the fragmented nature of this literature has stifled the widespread analysis of transients. We review the literature on transient analyses of linear PPM models and synthesise a coherent framework. We promote the use of standardised indices, and categorise indices according to their focus on either convergence times or transient population density, and on either transient bounds or case?specific transient dynamics. We use a large database of empirical PPM models to explore relationships between indices of transient dynamics. This analysis promotes the use of population inertia as a simple, versatile and informative predictor of transient population density, but criticises the utility of established indices of convergence times. Our findings should guide further development of analyses of transient population dynamics using PPMs or other empirical modelling techniques.</p

Transient analyses of population dynamics using matrix projection models

In this chapter we address the burgeoning topic of transient population dynamics using matrix projection models (MPMs), which are introduced in Chapter 9 and widely used for examining the dynamics of populations structured by discrete life cycle stages (e.g. developmental phases, sex) or discretized continuous life history traits (e.g. age, size; Caswell 2001).</p